Theorem sseqtrrd 3287

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2244	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3286	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  sseqtrrid  3299  fnfvima  5943  tfrlemiubacc  6591  tfr1onlemubacc  6607  tfrcllemubacc  6620  rdgivallem  6642  nnnninf  7456  nninfwlpoimlemg  7505  ccatass  11354  swrdval2  11401  dfphi2  12976  ctinf  13299  imasaddfnlemg  13612  imasaddvallemg  13613  subsubm  13767  subsubg  13977  subsubrng  14495  subsubrg  14526  lidlss  14785  toponss  15050  ssntr  15146  iscnp3  15227  cnprcl2k  15230  tgcn  15232  tgcnp  15233  ssidcn  15234  cncnp  15254  txcnp  15295  imasnopn  15323  hmeontr  15337  blssec  15462  blssopn  15509  xmettx  15534  metcnp  15536  plyaddlem1  15771  plymullem1  15772  plycoeid3  15781  nnsf  16953  nninfsellemsuc  16960